A168495 -id:A168495 - cp's OEIS frontend

A168494 Sequence with Hankel transform equal to 3^floor(n^2/3).

1, 1, 2, 7, 32, 160, 830, 4405, 23798, 130498, 724748, 4069258, 23064608, 131809108, 758696492, 4394825647, 25600773272, 149877922228, 881394158558, 5204245242208, 30841413359186, 183381577399006, 1093695670905206

Offset: 0

Paul Barry, Nov 27 2009

Hankel transform is A168495.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000108, A091866, A109033. - Philippe Deléham, Nov 27 2009

CoefficientList[Series[(1+x-Sqrt[1-10*x+25*x^2-12*x^3])/(6*x*(1-x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

G.f.: 1/(1-x/(1-x/(1-3x/(1-x/(1-x/(1-3x/(1-x/(1-x/(1-3x/(1-.... (continued fraction);
G.f.: 1/(1-x-x^2/(1-4x-3x^2/(1-2x-3x^2/(1-4x-x^2/(1-4x-3x^2/(1-2x-3x^2/(1-4x-x^2/(1-... (continued fraction),
with sequences (1,3,3,1,3,3,1,3,3,1,...) and (1,4,2,4,4,2,4,4,2,4,4,...).
G.f.: (1+x-sqrt(1-10x+25x^2-12x^3))/(6x(1-x)).
a(n) = Sum_{k=0..n} A091866(n,k)*3^(n-k). - Philippe Deléham, Nov 27 2009
Conjecture: (n+1)*a(n) +(4-11*n)*a(n-2) +5*(7*n-11)*a(n-2) +(92-37*n) * a(n-3) +6*(2*n-7)*a(n-4) = 0. - R. J. Mathar, Sep 30 2012
a(n) ~ sqrt(33-sqrt(33))*((7+sqrt(33))/2)^n/(12*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

A179533 Expansion of (1/(1-x-2x^2))*c(x/(1-x-2x^2)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 2, 7, 23, 85, 332, 1369, 5870, 25945, 117374, 540805, 2528675, 11966923, 57206972, 275824159, 1339721519, 6549093013, 32195473406, 159065828029, 789395034701, 3933239089903, 19668745466636, 98679891233803, 496570499905832, 2505670304785615, 12675395921692394, 64270076976110203, 326580624341708693, 1662796531746045157, 8481930651824392268, 43341418581113085697

Offset: 0

Paul Barry, Jan 08 2011

nonn

Hankel transform is A168495(n+1).

Cf. A000108, A073370, A168495.

with(LREtools): with(FormalPowerSeries): # requires Maple 2022
ogf:= (1/(1-x-2*x^2))*(1 - sqrt(1 - 4*(x/(1-x-2*x^2)))) / (2*(x/(1-x-2*x^2))):
init:= [1, 2, 7, 23, 85, 332, 1369];
iseq:= seq(u(i-1)=init[i],i=1..nops(init)): req:= FindRE(ogf,x,u(n));
rmin:= subs(n=n-4,MinimalRecurrence(req,u(n),{iseq})[1]); # Mathar's recurrence
a:= gfun:-rectoproc({rmin, iseq}, u(n), remember):
seq(a(n),n=0..30); # Georg Fischer, Nov 04 2022

G.f.: 1/(1-x-2x^2-x/(1-x/(1-x-2x^2-x/(1-x/(1-x-2x^2-x/(1-x/(1-x-2x^2-x/(1-x/(1-... (continued fraction);
a(n) = Sum_{k=0..n} A000108(k)*Sum_{j=0..n-k} C(k+j,k)*C(j,n-k-j)*2^(n-k-j).
a(n) = Sum_{k=0..n} A073370(n,k)*A000108(k).
D-finite with recurrence: (n+1)*a(n) +2*(1-3n)*a(n-1) +(n-1)*a(n-2) +4*(3n-5)*a(n-3) +4*(n-3)*a(n-4)= 0. - R. J. Mathar, Nov 17 2011

cp's OEIS Frontend

A168494 Sequence with Hankel transform equal to 3^floor(n^2/3).

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